6-moments couples and force couple systems
TRANSCRIPT
-
7/24/2019 6-Moments Couples and Force Couple Systems
1/69
Lecture #6
Moments, Couples, and Force
Couple Systems
-
7/24/2019 6-Moments Couples and Force Couple Systems
2/69
Equivalent Forces
We defined equivalent forces as being forces
with the same magnitude acting in the same
direction and acting along the same line of
action (this is through the Principle of
ransmissibility!, but why do the forces need
to act along the same line"
-
7/24/2019 6-Moments Couples and Force Couple Systems
3/69
4.1 Introduction to Moments
he tendency of a force to rotate a rigid body
about any defined a#is is called the Moment
of the force about the a#is
-
7/24/2019 6-Moments Couples and Force Couple Systems
4/69
MOMENT OF A FORE ! "ALAR FORMLATION $"ection 4.1%
he moment, M, of a force about a point provides a measure of the tendency for
rotation (sometimes called a torque!$
M
M % F & d
-
7/24/2019 6-Moments Couples and Force Couple Systems
5/69
Moment caused &' a Force
he Moment of Force (F! about an a#is
through Point ('! or for short, the Moment of
Fabout ', is the product of the magnitude of
the force and the perpendicular distance
between Point ('! and the line of action of
Force (F!
M'% Fd
-
7/24/2019 6-Moments Couples and Force Couple Systems
6/69
nits o( a Moment
he units of a Moment are Nmin the S) system
ftlbsor inlbsin the *S Customary system
-
7/24/2019 6-Moments Couples and Force Couple Systems
7/69
A))LIATION"
+eams are often used to bridge gaps in walls$ We have tonow what the effect of the force on the beam will have on
the beam supports$
What do you thin those impacts are at points ' and +"
-
7/24/2019 6-Moments Couples and Force Couple Systems
8/69
A))LIATION"
Carpenters often use a hammer in this way to pull a stubborn nail$hrough what sort of action does the force F- at the handle pull the nail"
-ow can you mathematically model the effect of force F- at point ."
-
7/24/2019 6-Moments Couples and Force Couple Systems
9/69
)ro*erties o( a Moment
Moments not only have
a magnitude, they also
have a sense to them$
he sense of a moment
is clocwise or counter/
clocwise depending
on which way it willtend to mae the ob0ect
rotate
-
7/24/2019 6-Moments Couples and Force Couple Systems
10/69
)ro*erties o( a Moment
he sense of a Moment is defined by the
direction it is acting on the '#is and can be
found using 1ight -and 1ule$
-
7/24/2019 6-Moments Couples and Force Couple Systems
11/69
+ari,non-s Teorem
The moment of a Force about any axis is
equal to the sum of the moments of its
components about that axis
his means that resolving or replacing forces
with their resultant force will not affect the
moment on the ob0ect being analy2ed
-
7/24/2019 6-Moments Couples and Force Couple Systems
12/69
MOMENT OF A FORE ! "ALAR FORMLATION (continued!
's shown, d is theperpendiculardistance from point . to the line of actionof
the force$
)n 3/4, the directionof M.is either clocwise or
counter/clocwise, depending on the tendency for rotation$
)n the 3/4 case, the magnitudeof the moment is Mo% F d
-
7/24/2019 6-Moments Couples and Force Couple Systems
13/69
REA/IN0 I2
5$ What is the moment of the 56 7 force about point '
(M'!"
'! 8 79m +! 8: 79m C! 53 79m
4! (53;8! 79m '
d % 8 m
F % 53 7
-
7/24/2019 6-Moments Couples and Force Couple Systems
14/69
E3am*le #1 ' 566/lb vertical force is applied to
the end of a lever which is attachedto a shaft at O$
4eterminea! Moment about O,
b! -ori2ontal force atA whichcreates the same moment,
c! Smallest force at ' whichproduces the same moment,
d! ?ocation for a 3@6/lb vertical
force to produce the samemoment,
e! Whether any of the forces fromb, c, and d is equivalent to theoriginal force$
-
7/24/2019 6-Moments Couples and Force Couple Systems
15/69
E3am*le #1
( )
( ) ( )in.12lb100
in.1260cosin.24
=
==
=
O
O
M
d
FdM
a) Moment about O is equal to the product of the
force and the perpendicular distance between the
line of action of the force and O. Since the force
tends to rotate the lever clockwise the moment
vector is into the plane of the paper.
inlb1200 =OM
-
7/24/2019 6-Moments Couples and Force Couple Systems
16/69
E3am*le #1
( )
( )
in.!.20
in.lb1200
in.!.20in.lb1200
in.!.2060sinin.24
=
==
==
F
F
FdM
d
O
b) "ori#ontal force atAthat produces the same
moment
lb$.%$=F
-
7/24/2019 6-Moments Couples and Force Couple Systems
17/69
E3am*le #1
( )
in.42
in.lb1200
in.42in.lb1200
=
=
=
F
F
FdMO
c) &he smallest force atAto produce the same
moment occurs when the perpendicular distance is
a ma'imum or whenFis perpendicular to OA.
lb%0=F
-
7/24/2019 6-Moments Couples and Force Couple Systems
18/69
E3am*le #1
( )
in.%cos60
in.%lb402
in.lb1200lb240in.lb1200
=
=
=
=
=
OB
d
d
FdMO
d) &o determine the point of application of a 240 lb
force to produce the same moment
in.10=OB
-
7/24/2019 6-Moments Couples and Force Couple Systems
19/69
E3am*le #1
e) (lthouh each of the forces in parts b) c) and d)
produces the same moment as the 100 lb force none
are of the same manitude and sense or on the same
line of action. *one of the forces is equivalent to the
100 lb force.
-
7/24/2019 6-Moments Couples and Force Couple Systems
20/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
21/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
22/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
23/69
4.4 )rinci*le o( Moments
AarignonBs heorem he moment of a force
about a point is equal to the sum of moments
of the components of the force about the
point
-
7/24/2019 6-Moments Couples and Force Couple Systems
24/69
MOMENT OF A FORE ! "ALAR FORMLATION (continued!
.ften it is easier to determine M.by using the components of Fas shown
(AarignonBs heorem!$
hen M.% (Fa! D (FEb!$ 7ote the different signs on the terms he typical
sign convention for a moment in 3/4 is that counter/clocwise is considered
positive$ We can determine the direction of rotation by imagining the body
pinned at . and deciding which way the body would rotate because of the
force$
For e#ample, M.
% F d and the
direction is counter/clocwise$
Fa
b
d
.
ab
.
F
F #
F y
-
7/24/2019 6-Moments Couples and Force Couple Systems
25/69
0RO) )ROLEM "OL+IN0
Since this is a 3/4 problem
5! 1esolve the 36 lb force along the
handleBs # and y a#es$
3! 4etermine M'using a scalar
analysis$
0iven5' 36 lb force is applied to the
hammer$Find5 he moment of the force at '$
)lan5
#
y
-
7/24/2019 6-Moments Couples and Force Couple Systems
26/69
0RO) )ROLEM "OL+IN0 $cont.%
"olution
G Fy % 36 sin 86H lb
G F# % 36 cos 86H lb
#
y
G M'% I(36 cos 86H!lb (5J in!(36 sin 36H!lb (K in!L
% 8K5$== lb9in % 8K3 lb9in (clocwise!
-
7/24/2019 6-Moments Couples and Force Couple Systems
27/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
28/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
29/69
Moments in /
4.7 Moment o( a Force a&out a
"*eci(ic A3is
)n 34 bodies the moment is due to a force
contained in the plane of action perpendicular
to the a#is it is acting around$ his maes the
analysis very easy$
)n 84 situations, this is very seldom found to
be the case$
-
7/24/2019 6-Moments Couples and Force Couple Systems
30/69
Moments in /
he moment about an a#is is still calculatedthe same way (by a force in the planeperpendicular to the a#is! but most forces are
acting in abstract angles$ +y resolving the abstract force into its
rectangular components (or into itscomponents perpendicular to the a#is of
concern! the moment about the a#is can thenbe found the same way it was found in 34 DM % Fd (where d is the distance between theforce and the a#is of concern!
-
7/24/2019 6-Moments Couples and Force Couple Systems
31/69
Notation (or Moments
)n simpler terms the Moment of a Force about
the y/a#is (My! can be found by using the
pro0ection of the Force on thex-zPlane
he 7otation used to denote Moments about
the Cartesian '#es are (Mx, My, and Mz!
-
7/24/2019 6-Moments Couples and Force Couple Systems
32/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
33/69
>iven the tension in cable +C is =66 7, find M#, My, and M2 about point '$
84 Moments
-
7/24/2019 6-Moments Couples and Force Couple Systems
34/69
Force ou*les
' Couple is defined as two Forces having the
same magnitude, parallel lines of action, and
opposite sense
)n this situation, the sum of the forces in each
direction is 2ero, so a couple does not affect
the sum of forces equations
' force couple will however tend to rotate thebody it is acting on
-
7/24/2019 6-Moments Couples and Force Couple Systems
35/69
Moment /ue to a Force ou*le
+y multiplying the magnitude of one Force by the
distance between the Forces in the Couple, the
moment due to the couple can be calculated$ M % Fdc
he couple will create a moment around an a#is
perpendicular to the plane that the couple falls in$
Pay attention to the sense of the Moment (1ight
-and 1ule!
-
7/24/2019 6-Moments Couples and Force Couple Systems
36/69
Moment o( a ou*le
&wo couples will have equal moments if
+ 2211 dFdF =
+ the two couples lie in parallel planes and
+ the two couples have the same sense orthe tendenc, to cause rotation in the same
direction.
-
7/24/2019 6-Moments Couples and Force Couple Systems
37/69
8' do 9e use Force ou*les:
he reason we use Force Couples to analy2e
Moments is that the location of the a#is the
Moment is calculated about does not matter
he Moment of a Couple is constant over the
entire body it is acting on
-
7/24/2019 6-Moments Couples and Force Couple Systems
38/69
ou*les are Free +ectors
he point of action of a Couple does notmatter
he plane that the Couple is acting in does
not matter'll that matters is the orientation of the plane
the Couple is acting in
herefore, a Force Couple is said to be aFree Aector and can be applied at any pointon the body it is acting
-
7/24/2019 6-Moments Couples and Force Couple Systems
39/69
Resolution o( +ectors
he Moment due to the Force Couple is
normally placed at the Cartesian Coordinate
.rigin and resolved into its #, y, and 2
components (Mx, My, and Mz!$
-
7/24/2019 6-Moments Couples and Force Couple Systems
40/69
+ector Addition o( ou*les
+y applying AarignonBs heorem to the Forces in the
Couple, it can be proven that couples can be added
and resolved as Aectors$
-
7/24/2019 6-Moments Couples and Force Couple Systems
41/69
Force ou*le "'stem
wo opposing force can be added to a rigid bodywithout affecting the equilibrium of it$
)f there is a force acting at a distance from an a#is,
two forces of equal magnitude and oppositedirection can be added at the a#is with out affectingthe equilibrium of the rigid body$
he original force and its opposing force at the a#ismae a couple that equates to a moment on therigid body$
he other force at the a#is results in the same forceacting on the body
-
7/24/2019 6-Moments Couples and Force Couple Systems
42/69
Force ou*le "'stems
's a result of this it can be stated that any
force (F) actin on a rii! bo!y may be
mo"e! to any i"en point on the rii! bo!y as
lon as a moment equal to moment of (F)about the axis is a!!e! to the rii! bo!y#
-
7/24/2019 6-Moments Couples and Force Couple Systems
43/69
Force ou*le "'stems
he reverse of this is also true, any force an!
moment actin at a point on a rii! bo!y can
be represente! as one force at a !istance
from the oriinal point $here the moment ofthe force about the oriinal point is equal to
the oriinal moment
R lt t ( " t (
-
7/24/2019 6-Moments Couples and Force Couple Systems
44/69
Resultant o( a "'stem o(
o*lanar Forces
iven a system of forces acting on a rigid body,
each force can be resolved into a force/couple
system acting at one point$
he Forces can then be added as Aectors and theMoments can be added as scalars as long as the
sense of each moment is taen into account$
his will reduce the system of forces to a single
force/moment system acting on the rigid body$ his can be reduced further to one force at a
distance Mr;Fr from the point of interest$
R l ti ( F i t
-
7/24/2019 6-Moments Couples and Force Couple Systems
45/69
Resolution o( a Force into a
Force ou*le "'stem in /
he same principles used in 34 apply in 84
also
he method used to resolve a force in 84 into
a Force/Couple system acting at one point is
to draw the rectangular coordinate system at
the point of interest$
1esolve the force in the plane it is in into theforce acting on one a#is and moment about
the a#is for each a#is one at a time$
R l ti ( " t (
-
7/24/2019 6-Moments Couples and Force Couple Systems
46/69
Resolution o( a "'stem o(
Forces in /
-
7/24/2019 6-Moments Couples and Force Couple Systems
47/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
48/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
49/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
50/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
51/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
52/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
53/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
54/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
55/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
56/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
57/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
58/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
59/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
60/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
61/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
62/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
63/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
64/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
65/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
66/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
67/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
68/69
-
7/24/2019 6-Moments Couples and Force Couple Systems
69/69